F77SB Introduction to Statistical Science B

Prof Jennie C Hansen

Course co-ordinator(s): Prof Jennie C Hansen (Edinburgh).

Aims:

The aims of this course are

  • To develop discrete probability models for data
  • To understand important features of these models

Summary:

This course provides an introduction to some of the probability models for inference. The main topics covered are:

  • Probability models: sample spaces and events; probability measures and their properties; simple probability calculations.
  • Conditioning and Independence: conditional probability; `Chain Rule’ for computing probabilities; Bayes’ Theorem; Partition Theorem; independence of events; probability calculations using concepts of conditioning and independence.
  • Random variables and their distributions, including the Binomial, Geometric, Poisson, Hypergeometric, and Indicator variables.
  • Expectation and standard deviation of a random variable: definitions and properties of expectation and standard deviation.

Detailed Information

Course Description: Link to Official Course Descriptor.

Pre-requisites: none.

Location: Edinburgh.

Semester: 2.

Syllabus:

1. Introduction to discrete probability models

  • Models for random experiments:
    • Sample spaces: events, set theory, Venn diagrams, de Morgan’s laws
    • Probability functions and the axioms of probability: Choice of probability function, consequences of the axioms.
  • Conditional probability and independence of events:
    • Definitions of P(A|B) and the definition of independence
    • `Chain rule’ for P(A intersection B)
    • Bayes Theorem
    • Partition Theorem and `tree diagrams’.

2. Special probability models for certain random experiments

  • Simple equally likely models
  • Sampling without replacement from a finite population:
    • ordered samples: model assumptions, calculations and relevant counting arguments
    • unordered samples: model assumptions, calculations and relevant counting arguments
  • Building models for a sequence of independent sub
  • experiments:
    • General model assumptions
    • Sampling with replacement: calculations and relevant counting arguments
    • Bernoulli trials and Binomial models: calculations and relevant counting arguments
    • Geometric models
    • Other examples

3. Discrete random variables

  • Definition of a discrete random variable
  • Probability mass functions for discrete random variables
  • Expectation and the properties of the expectation of a random variable.
  • Variance and the properties of the variance of a random variable.
  • Indicator variables

4. Special examples of discrete random variables.

  • Binomial and Bernoulli variables
  • Geometric and Negative Binomial variables
  • Poisson variables
  • Hypergeometric variables
  • Indicator variables

Learning Outcomes: Subject Mastery

At the end of this course, students should be able to

  • Carry out probability calculations for basic discrete probability models using standard techniques including the Partition Theorem, Bayes’ Theorem, and the Chain Rule.
  • Work out the distribution, expectation, and standard deviation for a discrete random variable that describes the outcome of a random experiment.
  • Identify an appropriate standard random variable to describe the outcome of a random experiment.
  • Use the properties of expectation and indicator variables to work out the expected value of a counting variable.

Reading list:

A First Course in Probability by S.M.Ross, 7th ed. (Pearson 2006).

Assessment Methods:

2-hour final exam in May (80%) and continuous assessment (2 assignments) (20%)

SCQF Level: 7.

Credits: 15.

Other Information

Help: If you have any problems or questions regarding the course, you are encouraged to contact the lecturer

VISION: further information and course materials are available on VISION